Face-Specific Growth and Dissolution Kinetics of Potassium

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Face-specific Growth and Dissolution Kinetics of Potassium Dihydrogen Phosphate Crystals from Batch Crystallization Experiments Holger Eisenschmidt, Andreas Voight, and Kai Sundmacher Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/cg501251e • Publication Date (Web): 11 Nov 2014 Downloaded from http://pubs.acs.org on November 16, 2014

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Crystal Growth & Design

Face-specific Growth and Dissolution Kinetics of Potassium Dihydrogen Phosphate Crystals from Batch Crystallization Experiments H. Eisenschmidt1, A. Voigt1, K. Sundmacher1,2* 1

Otto-von-Guericke University, Department of Process Systems Engineering, Magdeburg 2

Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg

Abstract: The final shape distribution of crystalline materials is an important product quality which is controlled by the growth and possibly also by the dissolution rates of individual crystal facets. Knowledge of the kinetics under batch process conditions enables optimal process design regarding desired shape distributions. In the present work, potassium dihydrogen phosphate (KDP) was chosen as model substance, for which face-specific growth and dissolution rates were determined in a batch crystallizer. The temporal evolution of the crystal population was tracked with a flow-through microscope using a shape estimation procedure presented in [1]. Effects of concentration and temperature on the kinetics were separately investigated. It was found that crystal growth is strongly affected by impurities at low supersaturation whereas impurity effects are diminishing at higher supersaturation. The dissolution rates were found to be linearly dependent on the applied undersaturation, and no impurity effects were visible. The temperature effects on both growth and dissolution kinetics were found to obey the Arrhenius law and corresponding activation energies for growth and dissolution of KDP were determined.

Corresponding author: Kai Sundmacher Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstraße 1, 39106 Magdeburg, Germany Tel: 0049 391 6110351 Fax: 0049 391 6110523 Email: [email protected]

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Face-specific Growth and Dissolution Kinetics of Potassium Dihydrogen Phosphate Crystals from Batch Crystallization Experiments H. Eisenschmidt1, A. Voigt1, K. Sundmacher1,2* 1

Otto-von-Guericke University, Department of Process Systems Engineering, Magdeburg 2

Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg

* Corresponding author: [email protected]

Abstract: The final shape distribution of crystalline materials is an important product quality which is controlled by the growth and possibly also by the dissolution rates of individual crystal facets. Knowledge of the kinetics under batch process conditions enables optimal process design regarding desired shape distributions. In the present work, potassium dihydrogen phosphate (KDP) was chosen as model substance, for which face-specific growth and dissolution rates were determined in a batch crystallizer. The temporal evolution of the crystal population was tracked with a flow-through microscope using a shape estimation procedure presented in [1]. Effects of concentration and temperature on the kinetics were separately investigated. It was found that crystal growth is strongly affected by impurities at low supersaturation whereas impurity effects are diminishing at higher supersaturation. The dissolution rates were found to be linearly dependent on the applied undersaturation, and no impurity effects were visible. The temperature effects on both growth and dissolution kinetics were found to obey the Arrhenius law and corresponding activation energies for growth and dissolution of KDP were determined.

Keywords: Crystallization, Crystal Shape, Faceted Crystals, Growth Kinetics, Dissolution Kinetics


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1.

Introduction

The size and shape distribution of crystalline materials strongly influences their solid state properties. For example needle-shaped and platelet-like crystals are known to cause difficulties during downstream processing operations such as filtration or drying and are therefore undesired in pharmaceutical applications [2]. However, due to the high prominence of specific facets on the crystal surface and a high surface to volume ratio, such crystal shapes are desirable in other applications, like catalysis [3] or solar cell applications [4]. The final size and shape distribution of crystalline materials is controlled by the underlying growth and dissolution kinetics [5,6]. Thus, control of the growth and dissolution kinetics opens a way for crystal shape control. This can for instance be achieved by changing of solvents, see for example Davey et al. [7], or by the application of special additives, which are typically used to decrease the growth rates of specific faces [8]. Another method for crystal shape control uses the supersaturation dependence of the relative face-specific growth and dissolution rates [9,10]. Importantly here, the crystal shape can be manipulated solely by supersaturation, i.e. by temperature control. In this way, the application of additional chemicals, which may alter the surface properties of the final product, can be omitted. The face-specific crystallization kinetics can, for instance, be determined by tracking the evolution of single crystals in a growth cell [9]. Since the crystals are typically imaged from a known and constant perspective [11], the geometrical state of the crystal can easily be tracked. Using this information the growth rates can then be deduced from this information. Since only single crystals can be observed with this technique, repeated experiments are required in order to avoid that the resulting kinetics are affected by phenomena like, for example, growth rate dispersion. But the application of the obtained kinetics to real process conditions is complicated. This is mainly due to fluid dynamic effects, particle-particle interactions or the presence of impurities in large scale processes. Hence, it is more desirable to determine the crystallization kinetics of an entire crystal population under real process conditions. This can be achieved by repeatedly taking samples of the crystal suspension during several crystallization runs [12]. These samples can then be analyzed offline using, for example, a standard microscope. However, by this method only a limited number of samples can be taken which is rather time consuming and requires careful sample preparation, i.e. filtration, washing and drying without altering the sampled crystals [13]. To circumvent these drawbacks, video microscopy has received much attention over the last two decades for the online monitoring of crystal suspensions [14]. This technology offers



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an immediate impression on the state of the crystal population. However, the shapes of the observed crystals, which are typically photographed with an unknown orientation [15], have to be reconstructed from the observed crystal projections. For this purpose several approaches have been presented in the open literature based on measured axis length distributions [15,16], generic crystal shape models [17], wire frame models [18] or Fourier descriptors [19]. Recently, Ochsenbein et al. [20] investigated the growth behavior of β L-Glutamic acid using video microscopy. For this purpose, several desupersaturation experiments were performed at different temperatures. Several kinetic approaches were fitted to the evolution of the measured length and width distribution of the crystal population, which was obtained by applying a generic crystal shape model. By this approach the authors were able to parameterize and predict the evolution of the crystal shape distribution. Due to the applied generic crystal shape model however, the face specific growth rates, particularly the growth rates of the {010} and {021} faces, could only be approximated. Borchert et al. [1] presented an estimation scheme in which the observed crystals are compared to a pre-computed database. The presented methods were applied to two batch cooling experiments of potassium dihydrogen phosphate (KDP) crystals and growth kinetics for the individual crystal faces were determined on the basis of an empirical power law. In the present work we extend these investigations by a detailed examination of the supersaturation and temperature dependence on the face-specific growth as well as the dissolution kinetics of KDP crystals. The article is organized as follows: In Section 2, we present the experimental setup for the determination of the face specific growth and dissolution rates. Furthermore, a simplified population balance model is presented which allows for the pre-computation of the applied temperature profiles. The results of this work are presented and discussed in Section 3. Finally, Section 4 concludes the work.

2.

Materials and Methods

2.1

Experimental

All experiments were performed in a flat-bottom 3L crystallization vessel without any additional vortex breakers. The solution was agitated by a four-blade pitch blade impeller (d = 4.5 cm) at a speed of 400 rpm. The solution temperature was measured using a PT100 thermometer. An ATR-FTIR probe, Nicolet iS10 Thermo Fisher, was installed in the vessel for online concentration monitoring. Each collected spectrum consisted of 16 independent


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scans in a range of 700 to 1800 cm-1 with a resolution 0.482 cm-1, resulting in a collection time of approx. 20 seconds. Spectra for various solution temperatures and concentrations were collected and used as calibration standards. During the crystallization experiments the FTIR spectra were automatically collected and the concentration is determined using the calibration spectra via a partial least squares approach. The state of the crystalline phase was monitored using a flow-through microscope (QicPic, Sympatec). This microscope was fed by an external sampling loop, which was driven by a peristaltic pump operated at a speed of 150-200 rpm. Videos of the crystal suspension were collected with a resolution of 1024 x 1024 pixels at a rate of 20 frames per second. The field of view had a size of 5000 µm by 5000 µm. These videos were processed after each experiment using the algorithms presented by Borchert et al. [1] to obtain the temporal evolution of the crystal size and shape distribution. The crystallizer temperature was controlled by two different thermostats which were connected to the vessel jacket by two tree way valves (see figure 1). This setup was chosen to enable rapid temperature changes to allow changes from growth to dissolution conditions, see section 3.2.

Figure 1: Schematic of the experimental setup KDP was purchased from Carl Roth GmbH + Co. KG (purity 98%, pH = 4.12 at Teq = 35 °C) and used without further purification. Seed crystals were obtained from Grüssing GmbH Analytica (purity 99.5%) and a sieve fraction from 212 µm to 300 µm was used in all experiments. For each experiment, a mass of KDP which corresponds to the desired supersaturation at the desired start temperature, see eq. (1), was dissolved in 1.8 kg deionized ACS Paragon Plus Environment


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water (conductivity 0.055 µS/cm). Once all crystals were dissolved, the solution was cooled down to the starting temperature. The seed crystals, 1.0 g for growth experiments and 0.8 g for dissolution experiments, were added to the solution as soon as this temperature was reached. At this point the video and data collection was started together with a program for temperature control (see section 2.3).

2.2

Potassium dihydrogen phosphate

Potassium dihydrogen phosphate (KDP) was chosen as model substance in this work. It crystallizes in the tetragonal space group I-42d with a = 7.460 Å and c = 6.982 Å [21]. KDP crystals have a well known shape consisting of prismatic {100} faces and pyramidal {101} faces (see figure 2). With these face types, the possible habits of a KDP crystal are ranging from rod-like crystals with a high prominence of the {100} faces on the outer crystal surface to compact crystals, as depicted in figure 2, and in an extreme case to octahedral crystals, exhibiting only the {101} faces on the outer crystal surface [22]. KDP has a rather high solubility in water. Its solubility is parameterized by the empirical curve: w eq = 4 .6479 ⋅ 10 − 5 T 2 − 0 .022596 T + 2 .8535

(1)

where T is the temperature in Kelvin and weq is the mass of KDP which is in equilibrium dissolved in 1 kg water. The parameters of this approach were determined based on our own gravimetric solubility measurements, which were found to be in good agreement with the values given by Mullin [23], see figure 3. The supersaturation can then be defined as

σ=

w −1 weq

(2)

where w is the actual mass of KDP dissolved in 1 kg water.

Figure 2: Left: Idealized geometry of a KDP crystal consisting of prismatic {100} faces (red) and pyramidal {101} faces (green), right: Representation of the crystal geometry by the face distances h1 and h2. ACS Paragon Plus Environment

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Figure 3: Solubility of KDP: Solid line: fitted second order polynomial eq. (1), red crosses: our measurements, blue circles: solubility data given by Mullin [23] 2.3

Supersaturation Control

Due to the possible variations in crystal habit (see section 2.2), the geometry of KDP crystals cannot be described with a one-dimensional framework. Here, we use the orthogonal distances h from the crystal surface to the center of mass to characterize the crystal shapes, see figure 2 right, where h1 denotes the distance of the {100} faces and h2 the distance to the {101} faces to the crystal center. The growth – or dissolution rates Gi are then defined as the displacement velocities of the {100} and {101} faces from the crystal center: Gi =

dhi dt

(3)

Using this framework and neglecting nucleation, agglomeration and breakage, the population balance for a batch crystallization of KDP reads: ∂f ∂f ∂f + G1 + G2 = 0, ∂t ∂ h1 ∂h2

(4)

where f is the number density in [#/m5]. The growth- or dissolution rate Gi of the i-th face is assumed to be independent on size and shape. An approximate solution of this population balance equation can be found by using the method of moments. These moments are in a twodimensional property space defined as [24]: ∞∞

µ i , j (t ) = ∫ ∫ h1i h2j f (h1 , h2 , t ) dh2 dh1

(5)

00

Using this definition, the population balance equation can be transformed to the following set of ordinary differential equations



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dµ 0,0 dt d µ 1, 0 dt dµ 0 ,1 dt

dµ i , j dt

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=0

(6a)

= G1 µ 0 , 0

(6b)

= G 2 µ 0,0

(6c)

= iG1 µ i −1, j + jG 2 µ i , j −1

(6d)

Although the temporal evolution of these moments yields a rather coarse approximation of the evolution of the crystal shape distribution, this approach suffices to predict the mass of solute which is consumed or released due to crystal growth or dissolution, respectively. Hence, the method of moments can be applied to derive a temperature control policy, to generate a constant supersaturation. Such operating policies have been developed for one dimensional population balance frameworks earlier [25]. In these frameworks, constant shape factors have to be applied that relate properties like surface area or volume to the actual crystal size. The shape of KDP crystals, however, is known to be strongly dependent on the applied supersaturation [1,10] and hence, the assumption of constant shape factors is not valid in this case. Applying temperature control policies based on such a one dimensional framework may therefore lead to poor results in the case of KDP. We therefore aim at deriving a more general framework for supersaturation control that accounts for the possible changes of KDP crystal shapes. The volume of a single KDP crystal can be calculated as [26] V cry ≈ 10 . 948 h12 h2 − 4 . 983 h13

(7)

Therefore, the mass consumption or release by growth or dissolution can be calculated by integrating eq. (7) over all crystals and the suspension volume V to give:

dµ 2,1 dµ3,0   dmKDP  = −ρ KDPV 10.948 − 4.983 dt dt dt  

(8)

At constant supersaturation, one needs to apply a temperature profile, which compensates this mass transfer, by changing the equilibrium concentration, such that the ratio w/weq remains constant. The total derivative of the supersaturation with respect to time reads:

dσ 1 dw w = − 2 dt weq dt weq

 ∂weq   ∂T 

 dT dmKDP 1 w  − 2  dt = w m dt weq eq H 2O 

 ∂weq   ∂T 

 dT   dt 

(9)

Clearly, the left hand side of eq. (9) is zero since the supersaturation has to remain constant. Inserting eqs. (1), (6) and (8) into (9) and rearranging finally yields:


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dT = dt

1 ρ KDPV (14.949 ⋅ G1µ 2,0 − 10.948 ⋅ (2G1µ1,1 + G2 µ 2,0 )) σ + 1 mH 2O 9.2958 ⋅ 10 −5 T − 0.022596

(10)

Eq. (10) is an ordinary differential equation that can be solved together with eqs. (6) yielding a temperature profile that has to be applied for a constant supersaturation. In the next section, the resulting temperature profiles of eq. (10) are applied to cooling crystallizations of KDP, to obtain the face specific growth and dissolution kinetics at constant levels of supersaturation and undersaturation respectively. However, to apply eq. (10), one already needs to know these growth and/or dissolution rates G = (G1,G2)T. Hence, these rates were approximately determined in preliminary experiments, results of which are not presented here.

3.

Results and Discussion

In this section, the obtained growth and dissolution kinetics are presented. To estimate the kinetics, only the evolution of the mean crystal shape of the seed crystal population was considered. For this purpose a classification between seed crystals and nucleated or agglomerated crystals is required. This classification was achieved by defining a rectangular region of a size of 150 µm x 150 µm which was initially centered at the mean seed crystal size T h seed,0 ≈ (122 µm, 140 µm ) . For each mean crystal shape estimate, 20 seconds of video were

processed, and all observed crystals within this region were considered as seed crystals. These crystal shape estimates were then subsequently used to estimate the current mean crystal shape and the center of rectangular seed crystal region was adjusted to coincide with this mean seed shape for the next iteration. With this approach the seed crystal evolution could be tracked reliably throughout the entire process time, and was further used for the estimation of the growth and dissolution kinetics, which is presented in the next sections.

3.1

Growth Kinetics

The general experimental procedure for the determination of the face-specific growth rates is depicted in figure 4. Once the seed crystals were added to the solution, the temperature of the crystallizer was controlled such that the supersaturation level remained constant during the experiment. Due to the increase in surface area which is available for growth, the temperature profile assumes a concave shape, see eq. (10). However, since the chosen seed loading was comparably small, the temperature was only decreased by 0.5 °C and thus, could be ACS Paragon Plus Environment


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considered as constant during the further evaluation of the experiments. It can be seen from figure 4, that the planned temperature profile can be followed well, and that the resulting supersaturation level is almost constant during the experiment. The mean crystal shape evolution is depicted on the right side of figure 4. Due to the constant growth conditions during the experiment, the slopes of the evolutions h1(t) and h2(t) are essentially constant and hence, no size or shape dependent growth is apparent. Face specific growth rates Gi, see eq. (3). have been obtained by applying a linear regression approach on the measured mean crystal shape evolution (see solid lines in figure 4, right).

Figure 4: Left: ideal (dashed line) and measured temperature (solid line) profiles. Middle: measured constant supersaturation profile. Right: Measured (marker) and regressed (solid lines) evolution of the mean seed crystal.

Figure 5: Growth kinetics of KDP crystals as a function of supersaturation at a temperature of T = 30°C. Red squares: growth rates of the prismatic {100} faces, green diamonds: growth rates of the pyramidal {101} faces. By repeating the experimental procedure described above for different supersaturation levels, growth kinetics, as exemplarily depicted in figure 5 for a set point temperature of 30 °C, were obtained. As can be seen, the relative growth rate


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Grel =

G1 G2

(11)

is, for all investigated supersaturations, lower the one, as the pyramidal {101} faces are always growing faster than the prismatic {100} faces. However, the relative growth rate Grel is constantly increasing with increasing supersaturation. Thus, the application of a low supersaturation level leads to the growth of elongated crystals, whereas the application of a high supersaturation leads to the formation of compact crystals, see figure 6, as already reported by Yang el at. [10] and Borchert et al. [1]. Below supersaturations of σ = 0.05, no significant growth was observable with our experimental setup, whose measurement error at these conditions was in the range of 1 nm/s. There is a strong increase of growth rates at a supersaturation between σ = 0.05 and σ = 0.08 and a linear dependency of the growth rates on supersaturation at levels above σ = 0.09, see figure 5. Such a behavior is well known for KDP crystals, see for example Sangwal [8] and Rashkovich et al. [27], in the presence of impurities. Especially the growth of the prismatic {100} faces is known to be strongly influenced by the presence of trivalent metal ions like Fe3+, Al3+ or Cr3+. In the presence of these ions, no growth takes place at low supersaturation while, after a transient supersaturation regime, the growth rates are linearly increasing at high supersaturation and independent on the impurity concentration [8,27,28]. These findings are clearly confirmed by our measurements.

Figure 6: Example figures of KDP crystals, left: seed crystals, middle: crystals grown at an intermediate supersaturation (σ = 0.074) and a temperature of T = 35°C, right: crystals grown at a high supersaturation (σ = 0.125) and a temperature of T = 35°C. Bars in the lower left corners correspond to a length of 200 µm. To further elucidate this behavior, we performed an additional set of growth experiments at a temperature of T = 35 °C where we dissolved KDP purchased from Grüssing with 99.5% purity, thus a higher purity as the previously used material from Roth ( ≥ 98%). The results for both materials are shown in figure 7. As can be seen, the change in KDP material has little influence on the growth behavior of the {101} faces. However, the death zone in which no significant growth of the {100} faces occurs is even more pronounced for the material purchased from Grüssing. At higher supersaturations, this effect is diminishing as the growth



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Figure 7: Supersaturation dependence of the face specific growth rates ({100}-red and {101}green). Dotted lines and empty markers correspond to KDP purchased from Roth, whereas KDP from Grüssing is represented by solid lines and filled markers. rates have approximately identical values. This indicates that the growth processes are controlled by fluid dynamics [2] rather than by the chemical composition of the solution at these supersaturation levels. To obtain kinetic parameters which are independent on the presence or absence of impurities, we considered only the linear growth regions, indicated by solid lines in figure 5 and 8, and hence the following kinetic approaches were parameterized for the set point temperatures of 25 °C, 30 °C, 35 °C, 40 °C and 45 °C:

(

G i = k g ,i σ − σ g* ,i

)

(12)

In eq.(12), kg,i is the growth rate constant and σ*g,i is a threshold supersaturation which is defined by the intersection on the linear equation (12) and the axis Gi = 0. Both parameters were determined for each set point temperature separately by linear regression. The resulting kinetics is shown together with all measured growth rates in figure 8. The same trends as already shown in figure 5 can be found for all temperatures. However, the absolute values of the growth rates increased significantly with increasing temperature. This can be attributed mainly to the temperature dependence of the growth rate constants kg,i which was described by the Arrhenius law (see figure 9):

 E A, g ,i k g ,i = k 0, g ,i exp −  RT

 .  

(13)

The activation energies EA,g,i for both face types, given in table 1, were determined and found to be in a similar range. From figure 8 it is also apparent that the threshold supersaturation σ*g,i is slightly decreasing with temperature, which is in accordance with the theoretical predictions given by Kubota [29] and Sangwal [8]. This temperature dependence was parameterized by the second order polynomial ACS Paragon Plus Environment

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σ *g ,i = s1,i T 2 + s 2 ,i T + s 3 ,i

(14)

where T is the temperature in °C and sj,i are empirical parameters listed in table 1. Table 1: Estimated parameters of eqs. (13) and (14) for the face specific growth rates of the {100} and {101} faces of KDP. {100} faces {101} faces 6 k0,g,i [µm/s] 6.01 x 10 13.7 x 106 EA,g,i [kJ/mol] 37.1 39.1 -5 s1,i [°C ²] 1.706 x 10 5.906 x 10-6 s2,i [°C-1] -1.913 x 10-3 -1.079 x 10-3 s3,i [-] 9.732 x 10-2 6.585 x 10-2

Figure 8: Obtained face-specific growth kinetics as function of supersaturation and temperature, left: {100} faces, right: {101} faces. Solid lines are indicating the supersaturation regions which were considered for the linear regression approach of eq. (12).

Figure 9: Left: Arrhenius plot of the growth rate constants kg,i as function of temperature right: temperature dependence of the threshold supersaturations σ*g,i defined by eq. (14) for the {100} faces (red, squares) and the {101} faces (green, diamonds) of KDP.



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3.2

Dissolution Kinetics

Figure 10 depicts a typical time-dependent dynamic dissolution experiment. Prior to the dissolution phase, the seed crystals were grown to reach a size that allowed for a sufficiently long dissolution time. These growth phases, phase I in figure 10, were realized using the same methods as described in the previous section. Once, the crystals had reached a sufficient size, the three way valves were switched to connect the second thermostat to the crystallizer jacket, see figure 1. This creates an undersaturated solution with minimal time effort (phase II in figure 10). When the desired level of undersaturation was reached, the solution temperature was controlled to maintain this undersaturation level using eq. (9) in phase III. The crystals were allowed to dissolve, until the mean seed crystal size of the prismatic faces reached a critical size, which was necessary to ensure that no seed crystals disappeared. Then, phase IV set in, where the solution was cooled down again in order to induce a second growth phase. Again, the mean evolution of the seed crystal population was tracked over the entire process time, see Figure 10 right. As can be seen, the mean crystal sizes were decreasing linearly over time during the dissolution phase, which indicates that the dissolution rates Di were neither size nor shape dependent. As in the case of the growth experiments, the dissolution rates Di for the given temperature and undersaturation could thus simply be determined by applying a linear regression approach to the measured mean shape evolution, see solid lines in figure 10 right, once the undersaturation level was constant. As soon as the solution was undersaturated, a fast disappearance of the crystal edges and vertices was observed, so that the crystals assumed an elliptical shape, see Figure 11 left. However, the same crystal shape model as for growing crystals was applied to analyze the recorded videos. The resulting shape estimates are exemplarily shown in Figure 11 right. It can be seen, that it is still possible to reproduce the mean crystal shape well with the applied shape model. A further indication, that the growth shape model is applicable is given by Figure 10. During the second growth phase the {100} and {101} faces are reappearing on the crystal surface and hence, the crystal shape model is adequate again. But no instantaneous jump in the mean shape evolution was visible when the process conditions changed from dissolution to growth modus, i.e. from phase III to phase IV. This would be a natural consequence of an inadequate shape model, which would be accompanied with a systematic error in the shape estimates. Hence, we conclude that the growth shape model of KDP is applicable for KDP crystals dissolving under the investigated conditions.


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The dissolution experiments were performed at the set point temperatures 30 °C, 35 °C, 40 °C and 45 °C and the resulting dissolution rates are depicted in figure 12. As can be seen, the dissolution rates Di are linearly dependent on the undersaturation and thus, a linear approach was chosen to parameterize the dissolution behavior:

(

D i = k d ,i σ − σ d* ,i

)

(15)

Figure 10: Left: ideal (dashed line) and measured temperature (solid line) profiles, Middle: resulting supersaturation profile. Right: Measured evolution of the mean seed crystal shape together with the regression results (solid lines) for the dissolution rate estimation.

Figure 11: Example frame of dissolving KDP crystals (left) together with the corresponding computer-based shape estimates [1] (right, color). In eq. (15), the parameters kd,i and σ*d,i were determined by linear regression. Figure 12 shows, that the parameters σ*d,i are somewhat lower than 0 as they vary between -0.002 to 0.01. Since this undersaturation range is affected by errors in the concentration measurement as well as uncertainties in the empirical solubility correlation (see eq. (2)) the effect of σ*d,i on the dissolution kinetics is not further discussed here, but used for the depiction of the resulting kinetics in figure 12. The dissolution rate constants kd,i show a significant temperature dependence, which was again described by an Arrhenius approach

 E A,d ,i   k d ,i = k0,d ,i exp − RT  

(16)

The estimated parameters of this approach are given in table 2. As can be seen in figure 12, both face types were dissolving with similar velocities regardless of the applied undersatuACS Paragon Plus Environment


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ration or temperatures. Also the dissolution rate constants were quite similar (see figure 13), resulting in similar activation energies. It can be noted that the dissolution proceeds generally faster compared to growth at similar driving forces (see figure 8). Furthermore, no significant effect of impurities on the dissolution behavior is apparent, which was clearly the case at growth conditions.

Figure 12: Obtained faces specific dissolution kinetics as a function of undersaturation and temperature, left: {100} faces, right: {101} faces.

Figure 13: Arrhenius plot of the dissolution rate constants kd,i as a function of temperature. Table 2: Estimated parameters of eq. (16) for the face-specific dissolution rates of the {100} and {101} faces of KDP. {100} faces {101} faces 6 k0,d,i [µm/s] 0.818 x 10 0.272 x 106 EA,d,i [kJ/mol] 29.7 26.8


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4.

Conclusions

In this article, the faceted growth and dissolution kinetics of KDP have been determined in batch crystallization experiments. The dependencies of the face specific growth rates on supersaturation reveal a significant influence of impurities present in the solution. However, these effects are diminishing at higher supersaturation levels, with indicates that batch-tobatch variations can be reduced under these conditions, since growth was found to be controlled by the transport of solute molecules from the bulk to the crystal surface. Dissolution was found to be generally faster than growth, and no effect of impurities was observed here. It was demonstrated, that the shape estimation procedure presented by Borchert et al. [1] is also applicable to the elliptical shapes of dissolving KDP crystals. Face-specific kinetics, determined under real process conditions, is the essential information for morphological population balance models which can be used to predict the evolution of the size and shape distributions. Hence, the here presented kinetics form the basis for advanced process control strategies towards crystal populations with desired final shape distribution, like for example model-based supersaturation control, which can be realized by pure temperature control. Since the {101} faces of KDP are growing almost exclusively at intermediate supersaturations whereas both the {100} and {101} faces grow with similar rates at high supersaturations, the final shape distribution of KDP crystal can be varied from elongated crystals to compact crystals, as exemplarily shown in figure 6. The application of growth-dissolution cycles may present another approach to crystal shape manipulation [30,31]. Since the {101} faces were found to grow faster than the {100} faces, whereas both faces dissolve with similar velocities, the application of growth-dissolution cycles will lead to elongated crystal shapes. Thus, the application of intermediate supersaturation levels, which may be accompanied with lattice defects and inhomogeneities [28], can be omitted by growthdissolution cycles.

Acknowledgements The authors gratefully acknowledge the support of this research work by the German Research Foundation (DFG) under the Grant SU 189/5-1.

Notation Greek ACS Paragon Plus Environment


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µ ρ σ

moment of a crystal distribution density relative supersaturation / undersaturation

Latin D EA f G G h k m R s T t V w

dissolution rate activation energy population density growth rate vector of growth rates face distance growth / dissolution rate constant mass universal gas constant empirical constant temperature time volume concentration

Subscripts cry d eq g KDP rel

crystal dissolution equilibrium growth Potassium dihydrogen phosphate relative

References [1]

Borchert, C.; Temmel, E.; Eisenschmidt, H. Lorenz, H.; Seidel-Morgenstern, A.; Sundmacher, K.: Image-Based in Situ Identification of Face Specific Crystal Growth Rates from Crystal populations. Crystal Growth & Design 2014, 14, 952-971

[2]

Lovette, M.A.; Browning, A.R.; Griffin, D.W.; Sizemore, J.P.; Snyder, R.C.; Doherty, M.F.: Crystal Shape Engineering. Ind. Eng. Chem. Res. 2008, 47, 9812-9833

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Yang, H.G.; Sun, C.H.; Qioa, S.Z.; Zou, J.; Liu, G.; Campbell Smith, S.; Chen, H.M.; Lu, G.Q.: Anatase TiO2 single crystals with a large percentage of reactive facets. Nature 2008, 453, 638-642

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Law, M.; Greene, L.E.; Johnson, J.C.; Saykally, R.; Yang, P.: Nanowire dye-sensitized solar cells. Nature Materials 2005, 4, 455-459

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Zhang, Y.; Sizemore, J.P.; Doherty, M.F.: Shape Evolution of 3-Dimensional Faceted Crystals. AIChE Journal 2006, 52, 1906-1915


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[10] Yang, G.; Kubota, N.; Sha, Z.; Louhi-Kultanen, M.; Wang, J.: Crystal Shape Control by Manipulating Supersaturation in Batch Cooling Crystallization. Crystal Growth & Design 2006, 6, 2799-2803 [11] Ma, C.Y.; Wan, J.; Wang, X.Z.: Faceted growth rate estimation of potash alum crystals grown from solution in a hot-stage reactor. Powder Technology 2012, 227, 96-103 [12] Ma, C. Y.; Wang, X. Z.: Model identification of crystal facet growth kinetics in morphological population balance modeling of L-glutamic acid crystallization and experimental validation. Chemical Engineering Science, 2012, 70, 22-30 [13] Puel, F.; Marchal, P.; Klein, J.: Habit transient analysis in industrial crystallization using two dimensional crystal sizing techniques. Chem. Eng. Res. Des. 1997, 75, 193205 [14] Patience, D. B.; Rawlings, J. B.: Particle-shape monitoring and control in crystallization processes. AIChE Journal. 2001, 47, 2125-2130 [15] Eggers, J.; Kempkes, M.; Mazzotti, M.: Measurement of size and shape distributions of particles through image analysis. Chemical Engineering Science, 2008, 63, 5513-5521 [16] Kempkes, M.; Vetter, T.; Mazzotti, M.: Measurement of 3D particle size distributions by stereoscopic imaging, Chemical Engineering Science, 2010, 65, 1362-1373 [17] Schorsch, S.; Vetter, T.; Mazzotti, M.: Measuring multidimensional particle size distributions during crystallization. Chemical Engineering Science, 2012, 77, 130-142 [18] Larsen, P. A.; Rawlings, J. B.; Ferrier, N. J.: Model-based object recognition to measure crystal size and shape distributions from in situ video images. Chemical Engineering Science, 2007, 62, 1430-1441 [19] Li, R. F.; Thomson, G. B.; White, G.; Wang, X. Z.; Calderon De Anda, J.; Roberts, K.: Integration of Crystal Morphology Modeling and On-Line Shape Measurement. AIChE Journal, 2006, 52, 2297-2305 [20] Ochsenbein, D.R.; Schorsch, S.; Vetter, T.; Mazzotti, M.; Morari, M.: Growth Rate Estimation of β L-Glutamic Acid from Online Measurements of Multidimensional Particle Size Distributions and Concentration. Ind. Eng. Chem. Res., 2014, 53, 91369148 [21] Botez, C.E.; Carbajal, D.; Adiraju, V.A.K.; Tackett, R.J.; Chianelli, R.R.: Intermediatetemperature polymorphic phase transition in KH2PO4: A synchrotron X-ray diffraction study. Journal of Physics and Chemistry of Solid, 2010, 71, 1576-1580 ACS Paragon Plus Environment


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[22] Borchert, C.; Sundmacher, K.: Efficient formulation of crystal shape evolution equations. Chemical Engineering Science, 2012, 84, 85-99 [23] Mullin, J.A.: Crystallization; Butterworth-Heinemann: Oxford, 2001 [24] Hulburt, H.M.; Katz, S.: Some problems in particle technology – A statistical mechanical formulation. Chemical Engineering Science, 1964, 19, 555-574 [25] Mullin, J.W.; Nyvlt, J.: Programmed cooling of batch crystallizers. Chemical Engineering Science, 1971, 26, 369-377 [26] Borchert, C.: Topics in Crystal Shape Dynamics, PhD Thesis, Otto-von-Guericke University Magdeburg, 2012 [27] Rashkovich, L.N.; Kronsky, N.V.: Influence of Fe3+ and Al3+ ions on the kinetics of steps on the {100} faces of KDP. Journal of Crystal Growth. 1997, 182, 434-441 [28] Zaitseva, N.; Carman, L.; Smolsky,I.; Torres, R.; Yan, M.: The effect of impurities and supersaturation on the rapid growth of KDP crystals. Journal of Crystal Growth, 1999, 204, 512-524 [29] Kubota, N.: Effect of impurities on the Growth Kinetics of Crystals. Cryst. Res. Technol. 2001, 36, 749-769 [30] Bajcinca, N.: Time optimal cyclic crystallization. Computers and Chemical Engineering, 2013, 58, 381-389 [31] Lovette, M. A.; Muratore, M.; Doherty, M. F.: Crystal Shape Modification Through Cycles of Dissolution and Growth: Attainable Regions and Experimental Validation. AIChE Journal, 2012, 58, 1465-1474


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For Table of Contents Use Only Title: Face-specific Growth and Dissolution Kinetics of Potassium Dihydrogen Phosphate Crystals from Batch Crystallization Experiments Authors: Holger Eisenschmidt, Andreas Voigt, Kai Sundmacher Synopsis: The face specific growth and dissolution rates of potassium dihydrogen phosphate crystals are determined as a function of supersaturation and temperature. All experiments were performed at constant super-/ undersaturation and approximately constant temperature. Hence, the influence of these two important conditions on the crystallization kinetics could be investigated and parameterized separately. TOC graphic


Face-Specific Growth and Dissolution Kinetics of Potassium

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